For an IVP $y' = f(x,y)$, with initial condition $y(x_0) = \alpha$, can something be said about how large the error at $x_1$, $x_1 > x_0$, is going to be if instead we had started with the initial condition $y(x_0) = \alpha + h$ for a small $h$? Of course, I'm assuming both solutions guaranteed by the Existence-Uniqueness theorem extend to $x_1$.
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Yes, one can say pretty much (assuming enough regularity). The buzzword is "dependence on the initial value" and you may have a look at this for example.